Multi-time correlations in the positive-P, Q, and doubled phase-space representations

Abstract

A number of physically intuitive results for the calculation of multi-time correlations in phase-space representations of quantum mechanics are obtained. They relate time-dependent stochastic samples to multi-time observables, and rely on the presence of derivative-free operator identities. In particular, expressions for time-ordered normal-ordered observables in the positive-P distribution are derived which replace Heisenberg operators with the bare time-dependent stochastic variables, confirming extension of earlier such results for the Glauber-Sudarshan P. Analogous expressions are found for the anti-normal-ordered case of the doubled phase-space Q representation, along with conversion rules among doubled phase-space s-ordered representations. The latter are then shown to be readily exploited to further calculate anti-normal and mixed-ordered multi-time observables in the positive-P, Wigner, and doubled-Wigner representations. Which mixed-order observables are amenable and which are not is indicated, and explicit tallies are given up to 4th order. Overall, the theory of quantum multi-time observables in phase-space representations is extended, allowing non-perturbative treatment of many cases. The accuracy, usability, and scalability of the results to large systems is demonstrated using stochastic simulations of the unconventional photon blockade system and a related Bose-Hubbard chain. In addition, a robust but simple algorithm for integration of stochastic equations for phase-space samples is provided.

On the survival of the quantum depletion of a condensate after release from a magnetic trap

“…Indeed, such a reduction of depletion was observed, strongly supporting the hypothesis that adiabaticity is needed for agreement with the results of [33]. More detailed study of this will be reported in [68].…”

“…However, over adiabatic timescales, this still introduced far too much noise in the Bogoliubov fields ψ B , ψ B to be useful. fact that an instantaneous quantum quench of a weakly interacting condensate produces depletion with a similar time-integrated momentum profile dt n B (k, t) ∝ 1/k 4 to the ground state value, but with a somewhat higher depletion, as described in detail in [68,78]. Simulations show that non-instantaneous but rapid quenches produce lower amounts of quantum depletion, as shown in Fig.…”

“…Therefore a theory that explicitly imposes orthogonality between condensate and Bogoliubov modes is required. We summarise it below, while details of the derivation and proper benchmarking of the method will be reported in [68].…”

“…Indeed, as expected we find that the survival of depleted atoms is enhanced in the anisotropic trap compared to the spherical one. Details will be reported in [68].…”

On the survival of the quantum depletion of a condensate after release from a magnetic trap

“…We used the robust stochastic integration procedure described in Ref. 82 . The and are independent white Gaussian noise fields of zero mean and variance: …”